Convexity and measures of statistical association

Recent investigations on the measures of statistical association highlight essential properties such as zero-independence (the measure is zero if and only if the random variables are independent), monotonicity under information refinement, and max-functionality (the measure of association is maximal if and only if we are in the presence of a deterministic (noiseless) dependence). An open question concerns the reasons why measures of statistical associations satisfy one or more of those properties but not others. We show that convexity plays a central role in all properties. Convexity plus a form of strictness (that we are to define) are necessary and sufficient for zero-independence, and convexity and strict convexity on Dirac masses are necessary and sufficient for max-functionality. We apply the findings to study the families of measures of statistical association based on Csisz & aacute;r divergences, optimal transport, kernels, as well as Chatterjee's new correlation coefficient. We further discuss the role of convexity in guaranteeing the asymptotic unbiasedness of given data estimators, prove a central limit theorem for those estimators under independence, and show the rate of convergence under arbitrary dependence. We demonstrate the findings with numerical simulations in a multivariate response context.